ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4776 on D5719: Exponential moment
Hoeffding's lemma
Formulation 0
Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that
(i) $[a, b] \subset \mathbb{R}$ is a D544: Closed real interval
(ii) \begin{equation} a < b \end{equation}
(iii) \begin{equation} \mathbb{P}(X \in [a, b]) = 1 \end{equation}
Let $\lambda \in \mathbb{R}$ be a D993: Real number.
Then \begin{equation} \mathbb{E} \left( e^{\lambda (X - \mathbb{E} X)} \right) \leq \exp \left( \frac{\lambda^2 (b - a)^2}{8} \right) \end{equation}